If the CSA had won the war the animosity would have died sooner than it did with the USA winning.
The CSA winning the war would simply mean that the USA had to accept the fact that there would be two separate countries. The CSA had no desire to invade the USA. The Supreme Court would have eventually ruled that secession was unconstitutional so no one would attempt it in the future. The USA may have even amended its constitution to explicitly prohibit secession.
By the end of the century relations would have normalized. We're friends with Vietnam, aren't we?
For the USA to win it had to militarily defeat and occupy the CSA.